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Integration is a fundamental concept in calculus, which is all about finding the antiderivative of a function. When you integrate, you are reversing differentiation. This process returns a function whose derivative gives the original function.
When looking at our specific example, we integrate the function \( u(r) = e^{-2r} \). To find its antiderivative, we use the formula for exponential functions:

• For a function of the form \( e^{ax} \), the integral is \( \frac{1}{a} e^{ax} + C \), where \( C \) is an arbitrary constant.

This constant \( C \) represents an infinite family of functions, showing that there are many functions which can yield the same derivative.
For \( u(r) \), applying this rule results in the antiderivative \( v(r) = -\frac{1}{2}e^{-2r} + C \). This is the most general form of our function, encompassing all possible solutions.

Differentiation is the reverse process of integration. It involves finding the derivative of a function, which tells us the rate of change or slope of the function at any point. In simpler terms, it breaks things down into their rate parts.
In order to verify our antiderivative, we differentiated our result, \( v(r) = -\frac{1}{2}e^{-2r} + C \), with respect to \( r \). Here's why it works:

• The derivative \( \frac{d}{dr}(-\frac{1}{2}e^{-2r}) \) involves multiplying by the derivative of the inside \( -2 \), thereby canceling out the \( -\frac{1}{2} \), resulting back in \( e^{-2r} \).
• The derivative of a constant \( C \) is 0, which does not affect the rest of the derivative calculation.

By performing differentiation, we confirm that we return to the original function \( u(r) \). This ensures our integration process and the calculated antiderivative are correct.

Exponential functions are a special kind of mathematical function often seen in many areas of science and finance. The function has the form \( f(x) = e^{ax} \), where \( a \) is a constant. One unique characteristic of exponential functions is that they change at rates proportional to their current value.
In mathematical calculations, such as integration and differentiation, exponential functions hold unique properties. The main characteristic when integrating exponential functions is that their form remains relatively unchanged, which simplifies calculations.
When differentiating an exponential function like \( e^{-2r} \), the result is still an exponential function. Specifically, differentiation involves applying the chain rule, which scales the original function’s exponential term. This consistent form makes exponential functions straightforward to manipulate, maintaining simplicity across many calculus operations.